Chemistry 232 Kinetics of Complex Reactions
Mar 31, 2015
Chemistry 232
Kinetics of Complex Reactions
The Pre-Equilibrium Approximation
Examine the following process1
1
2
1 1
1 1
2 2
v k A
v k B
v k B
k
k
k
A B
B C
2
d Ck B
dt
Pre-Equilibrium (II)
B is obviously an intermediate in the above mechanism. • Could use SSA.
What if the initial equilibrium is fast?• Step 2 is the rds!
1
1
B A Ak
Kk
Pre-Equilibrium (III)
We now have a simple expression for the [B]; hence
12 2
1
A ' Ad C k
k B k kdt k
Lindemann-Hinshelwood Mechanism
An early attempt to explain the kinetics of complex reactions.
PA
A2AA
AAAA
2
1
1
k
k
k
Akv
AAkv
Akv
22
11
211
Mechanism Rate Laws
The ‘Activated’ Intermediate
Formation of the product depends directly on the [A*].
Apply the SSA to the net rate of formation of the intermediate [A*]
0AkAAkAkdtAd
212
1
Is That Your ‘Final Answer’?
Substituting and rearranging
Akk
AkkdtPd
12
212
The ‘Apparent Rate Constant’ Depends on Pressure
The rate laws for the Lindemann-Hinshelwood Mechanism are pressure dependent.
High Pressure Case Low Pressure Case
Ak
kAkk
dtPd
1
12
/
2
21
Ak
AkdtPd
/
The Pressure Dependence of k’ In the Lindemann-Hinshelwoood
Mechanism, the rate constant is pressure dependent.
21
1
1 kkk
Ak1
k1 /
Catalysts
So far, we have considered one way of speeding up a reaction (i.e. increasing T usually increases k). Another way is by the use of a catalyst.
A catalyst - a substance that speeds up the rate of the reaction without being consumed in the overall reaction.
look at the following two reactions
A+B C rate constant k
A+B C rate constant with catalyst is kcat
NOTE: RATE WITH CATALYST > RATE WITHOUT CATALYST
Types of Catalyst
We will briefly discuss three types of catalysts. The type of catalyst depends on the phase of the catalyst and the reacting species. • Homogeneous
• Heterogeneous
• Enzyme
Homogeneous Catalysis
The catalyst and the reactants are in the same phase
e.g. Oxidation of SO2 (g) to SO3 (g)
2 SO2(g) + O2(g) 2 SO3 (g) SLOW
Presence of NO (g), the following occurs.NO (g) + O2 (g) NO2 (g)
NO2 (g) + SO2 (g) SO3 (g) + NO (g) FAST
SO3 (g) is a potent acid rain gas
H2O (l) + SO3 (g) H2SO4 (aq) Note the rate of NO2(g) oxidizing SO2(g)
to SO3(g) is faster than the direct oxidation.
NOx(g) are produced from burning fossil fuels such as gasoline, coal, oil!!
Heterogeneous Catalysis
The catalyst and the reactants are in different phases• adsorption the binding of molecules on a
surface.
Adsorption on the surface occurs on active sites• Places where reacting molecules are adsorbed
and physically bond to the metal surface.
The hydrogenation of ethene (C2H4 (g)) to ethane
C2H4 (g) + H2(g) C2H6 (g)Reaction is energetically favourablerxnH = -136.98 kJ/mole of ethane.
With a finely divided metal such as Ni (s), Pt (s), or Pd(s), the reaction goes very quickly .
There are four main steps in the process
• the molecules approach the surface;
• H2 (g) and C2H4 (g) adsorb on the surface;
• H2 dissociates to form H(g) on the surface; the adsorbed H atoms migrate to the adsorbed C2H4 and react to form the product (C2H6) on the surface
• the product desorbs from the surface and diffuses back to the gas phase
Simplified Model for Enzyme Catalysis
E enzyme; S substrate; P product
E + S ES ES P + E
rate = k [ES]The reaction rate depends directly on the
concentration of the substrate.
Enzyme Catalysis
Enzymes - proteins (M > 10000 g/mol)High degree of specificity (i.e., they will
react with one substance and one substance primarily
Living cell > 3000 different enzymes
The Lock and Key Hypothesis
Enzymes are large, usually floppy molecules. Being proteins, they are folded into fixed configuration.
According to Fischer, active site is rigid, the substrate’s molecular structure exactly fits the “lock” (hence, the “key”).
The Lock and Key (II)
The Michaelis-Menten Mechanism
Enzyme kinetics – use the SSA to examine the kinetics of this mechanism.
ES – the enzyme-substrate complex.
1 2
1
k k
kE S ES P E
Applying the SSA to the Mechanism
Note that the formation of the product depends directly on the [ES]
What is the net rate of formation of [ES]?
ESkESkSEkdtESd
21o1
ES – The Intermediate
Apply the SSA to the equation for d[ES]/dt = 0
21
o1
21o1
kkSEk
ES
ESkESkSEk
0dtESd
Working Out the Details
Let [E]o = [E] + [ES]
Initial enzyme concentration
Complex concentration
Free enzyme concentration
o121
oo1
SkkkSEk
ES
Note that [E] = [E]o - [ES]
The Final Equation
Substituting into the rate law vp.
ESkv 2p
Mo
oo2
o121
oo12p
KSSEk
SkkkSEk
kv
The Michaelis Constant and the Turnover Number
The Michaelis Constant is defined as
1
12M k
kkK
The rate constant for product formation, k2, is the turnover number for the catalyst.
Ratio of k2 / KM – indication of catalytic efficiency.
The Maximum Velocity
As [S]o gets very large.
maxlim vEkv o2
Sp
o
Note – Vmax is the maximum velocity for the reaction. The limiting value of the reaction rate high initial substrate concentrations.
Lineweaver-Burk Equation
Plot the inverse of the reaction rate vs. the inverse of the initial substrate concentration.
oM
o S1
vK
v1
v1
maxmax
Chain Reactions
Classifying steps in a chain reaction.• Initiation
• C2H6 (g) 2 CH3•
• Propagation Steps
• C2H6 + •CH3 •C2H5 + CH4
• Branching Steps
• H2O + •O• 2 •OH
Chain Reactions (Cont’d)
Retardation Step
• HBr + H• H2 + Br•
Terminations Steps
• 2 CH3CH2• CH3CH2CH2CH3
Inhibition Steps
• R• + CH3• RCH3
The H2 + Br2 Reaction
The overall rate for the reaction was established in 1906 by Bodenstein and Lind
HBrkBr
BrHk
dt
HBrd/
2
23
22
The Mechanism
The mechanism was proposed independently by Christiansen and Herzfeld and by Michael Polyani.
Mechanism
Rate Laws
BrBr 22
HHBrHBr 2
BrHBrBrH 2
BrHHBrH 2
2BrBrBr
211 Brkv 222 HBrkv HBrkv 222 HBrHkv 33
244 Brkv
Using the SSA
Using the SSA on the rates of formation of Br• and H•
HBrkkBr
BrHkkk
dt
HBrd
/2
32
23
224
122
Hydrogenation of Ethane
The Rice-Herzfeld Mechanism
Mechanism 362 2CHHC
423362 CHCHCHCHHC HCHCHCHCH 2223
22333 HCHCHHCHCH 6223 HCHCHCH
Rate Laws for the Rice-Herzfeld Mechanism
The rate laws for the elementary reactions are as follows.
6211 HCkv
36222 CHHCkv
2322 CHCHkv 3322 // CHCHHkv
2333 CHCHHkv
Explosions
Thermal explosions • Rapid increase in the reactions rate with
temperature.
Chain branching explosions• chain branching steps in the mechanism lead
to a rapid (exponential) increase in the number of chain carriers in the system.
Photochemical Reactions
Many reactions are initiated by the absorption of light.
Stark-Einstein Law – one photon is absorbed by each molecule responsible for the primary photochemical process.
I = Intensity of the absorbed radiation
Iv I
Primary Quantum Yield
Define the primary quantum yield,
absorbed photons of #
productsprimary of #
Define the overall quantum yield,
absorbed photons of #
react that molecules reactant of #
Photosensitization
Transfer of excitation energy from one molecule (the photosensitizer) to another nonabsorbing species during a collision..
HHgHHHg
HHgHHg
HgHgnm
2
2
254
2
Polymerization Kinetics
Chain polymerization• Activated monomer attacks another
monomer, chemically bonds to the monomer, and then the whole unit proceeds to attack another monomer.
Stepwise polymerization• A reaction in which a small molecule (e.g.,
H2O) is eliminated in each step.
Chain Polymerization
The overall polymerization rate is first order in monomer and ½ order in initiator.
The kinetic chain length, kcl
• Measure of the efficiency of the chain propagation reaction.
produced centres active of #
consumed units monomer of #
i
pkcl v
v
Mechanism
InitiationI 2 R•
Or
M + R• M1 • Propagation
M + M1• M2 •
M + M2• M3 •
M + M3• M4 •Etc.
Ikv ii
Rate Laws
1npp MMkv
Mechanism (Cont’d)
Termination
M + M3• M4 •
2 Mkv tt
Note – Not all the initiator molecules produce chainsDefine = fraction of initiator molecules that produce chains
Ikdt
Mdi2
Return to Kinetic Chain Length
We can express the kinetic chain length in terms of kt and kp
2
12
12
2
2
pkcl
t
p
i t
k M M
k M
k M I
k k
Stepwise Polymerization
A classic example of a stepwise polymerization – nylon production.NH2-(CH2)6-NH2 + HOOC-(CH2)4COOH
NH2-(CH2)6-NHOC-(CH2)4COOH + H2O
After many stepsH-(NH-(CH2)6-NHOC-(CH2)4CO)n-OH
The Reaction Rate Law
Consider the condensation of a generic hydroxyacid
OH-M-COOH
Expect the following rate law
COOHOHkv polypoly
The Reaction Rate Law (Cont’d)
Let [A] = [-COOH]A can be taken as any generic end
group for the polymer undergoing condensation.
Note 1 –OH for each –COOH
2Ak
AOHkv
poly
polypoly
The Reaction Rate Law (Cont’d)
If the rate constant is independent of the molar mass of the polymer
opoly
o
opoly
ot
Atk
A
COOHtk
COOHCOOH
1
1
The Fraction of Polymerization
Denote p = the fraction of end groups that have polymerized
o
to
A
AAp
opoly
opoly
Atk
Atkp
1
Statistics of Polymerization
Define Pn = total probability that a polymer is composed of n-monomers
ppP nn 11
The Degree of Polymerization
Define <n> as the average number of monomers in the chain
t
o
A
A
pn
1
1
Degree of Polymerization (cont’d)
The average polymer length in a stepwise polymerization increases as time increases.
opoly
opoly
opoly
Atk
Atk
Atk
pn
1
11
1
1
Molar Masses of Polymers
The average molar mass of the polymer also increases with time.
Two types of molar mass distributions.
• <M>n = the number averaged molar mass of the polymer.
• <M>w = the mass averaged molar mass of the polymer.
Definitions of <M>n
Two definitions!
11
1
on
J JJ
M Mp
n Mn
Mo = molar mass of monomer n = number of polymers of mass Mn
MJ = molar mass of polymer of length nJ
Definitions of <M>w
<M>w is defined as follows
J JJ
JJ
j
xnow
Mn
Mn
pxMpM n 1221
Note - xn the number of monomer units in a polymer molecule
The Dispersity of a Polymer Mixture
Polymers consists of many molecules of varying sizes.
Define the dispersity index () of the mass distribution.
n
w
M
M
Note – monodisperse sample ideally has <M>w=<M>n
The Dispersity Index in a Stepwise Polymerization
The dispersity index varies as follows in a condensation polymerization
n
w
M
M1
Note – as the polymerization proceeds, the ratio of <M>w/<M>n approaches 2!!!
Mass Distributions in Polymer Samples
For a random polymer sample
09 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41
Monodisperse Sample
Polydisperse Sample
Molar mass / (10000 g/mole)
Pn